Clusterring and Classification

Preparing environment

library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.3     ✔ readr     2.1.4
## ✔ forcats   1.0.0     ✔ stringr   1.5.0
## ✔ ggplot2   3.4.4     ✔ tibble    3.2.1
## ✔ lubridate 1.9.3     ✔ tidyr     1.3.0
## ✔ purrr     1.0.2     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(dplyr)
library(ggplot2)
library(corrplot)
## Warning: package 'corrplot' was built under R version 4.3.2
## corrplot 0.92 loaded
date()
## [1] "Sun Nov 26 16:16:06 2023"

Task 2 - Loading and exploring the MASS data package

library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
data("Boston")
colnames(Boston)
##  [1] "crim"    "zn"      "indus"   "chas"    "nox"     "rm"      "age"    
##  [8] "dis"     "rad"     "tax"     "ptratio" "black"   "lstat"   "medv"
dim(Boston)
## [1] 506  14
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
head(Boston)
##      crim zn indus chas   nox    rm  age    dis rad tax ptratio  black lstat
## 1 0.00632 18  2.31    0 0.538 6.575 65.2 4.0900   1 296    15.3 396.90  4.98
## 2 0.02731  0  7.07    0 0.469 6.421 78.9 4.9671   2 242    17.8 396.90  9.14
## 3 0.02729  0  7.07    0 0.469 7.185 61.1 4.9671   2 242    17.8 392.83  4.03
## 4 0.03237  0  2.18    0 0.458 6.998 45.8 6.0622   3 222    18.7 394.63  2.94
## 5 0.06905  0  2.18    0 0.458 7.147 54.2 6.0622   3 222    18.7 396.90  5.33
## 6 0.02985  0  2.18    0 0.458 6.430 58.7 6.0622   3 222    18.7 394.12  5.21
##   medv
## 1 24.0
## 2 21.6
## 3 34.7
## 4 33.4
## 5 36.2
## 6 28.7
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
The dataframe describes 506 towns in the Boston area across 14 variables.
library(reshape)
## 
## Attaching package: 'reshape'
## The following object is masked from 'package:lubridate':
## 
##     stamp
## The following object is masked from 'package:dplyr':
## 
##     rename
## The following objects are masked from 'package:tidyr':
## 
##     expand, smiths
Boston_melted <- melt(Boston)
## Using  as id variables
ggplot(Boston_melted, aes(x = value)) +
  geom_histogram(binwidth = 1, fill = "turquoise", color = "blue") +
  facet_wrap(~variable, scales = "free") +
  theme_minimal()

\(~\)

Here looking at the data as a distribution we can begin to see some issues. Variables black, crime and age seem to be skewed. Most of the houses seem to have 6 rooms, which is quite lovely.

\(~\)

Task 3 - Graphical overview of data and pairwise correlations

cor_matrix <- cor(Boston) 
corrplot(cor_matrix, method="circle")

\(~\) ##### Here I can see there are some strong negative corerelations: dis v (indus-nox-age) and mdev v lstat. Also there are some strong positive correlations: mdev v rm (which makes sense as the more rooms you have, ideally, the higher the value of your property), idus v nox (the more industry you have the higher the relative air pollution level).

\(~\)

In looking at this data set I am really intersted, at least at first, by what crime correlates too. Here I can see that it positively correlates to rad and tax and negatively coreelates to dis, black and mdev. I could interpret this as crime being correlated to more rural areas and that the higher the mediam value of the properties in your town the lower the crime rate which also makes sense.

\(~\)

Task 4 - Standardize the data prior to LDA

Scaling the Boston data
Boston_scaled <- as.data.frame(scale(Boston))
summary(Boston_scaled)
##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865
bins <- quantile(Boston_scaled$crim)

# create a categorical variable 'crime'
crime <- cut(Boston_scaled$crim, breaks = bins,labels = c("low","med_low","med_high","high"), include.lowest = TRUE)

# remove original crim from the dataset
Boston_scaled <- dplyr::select(Boston_scaled, -crim)

# add the new categorical value to scaled data
Boston_scaled <- data.frame(Boston_scaled, crime)

summary(Boston_scaled)
##        zn               indus              chas              nox         
##  Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723   Min.   :-1.4644  
##  1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723   1st Qu.:-0.9121  
##  Median :-0.48724   Median :-0.2109   Median :-0.2723   Median :-0.1441  
##  Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723   3rd Qu.: 0.5981  
##  Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648   Max.   : 2.7296  
##        rm               age               dis               rad         
##  Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658   Min.   :-0.9819  
##  1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049   1st Qu.:-0.6373  
##  Median :-0.1084   Median : 0.3171   Median :-0.2790   Median :-0.5225  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617   3rd Qu.: 1.6596  
##  Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566   Max.   : 1.6596  
##       tax             ptratio            black             lstat        
##  Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033   Min.   :-1.5296  
##  1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049   1st Qu.:-0.7986  
##  Median :-0.4642   Median : 0.2746   Median : 0.3808   Median :-0.1811  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332   3rd Qu.: 0.6024  
##  Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406   Max.   : 3.5453  
##       medv              crime    
##  Min.   :-1.9063   low     :127  
##  1st Qu.:-0.5989   med_low :126  
##  Median :-0.1449   med_high:126  
##  Mean   : 0.0000   high    :127  
##  3rd Qu.: 0.2683                 
##  Max.   : 2.9865
Just checking that the code worked and I now have a crime categorical variable and crim has been dropped.
# Test and train data sets:
n <- nrow(Boston_scaled)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create train set
train <- Boston_scaled[ind,]

# create test set 
test <- Boston_scaled[-ind,]
Here we are preparing the data into test and train sets. We randoming choose 80 percent of the data into df ind which was used in train and the remainder or -ind was used in test.

\(~\)

Task 5 - Fitting Linear Discriminate analysis

# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2549505 0.2425743 0.2524752 0.2500000 
## 
## Group means:
##                   zn      indus        chas        nox          rm        age
## low       0.95434742 -0.9258605 -0.11943197 -0.8741718  0.45741131 -0.8706537
## med_low  -0.08231376 -0.2915812 -0.03128211 -0.5969715 -0.10157606 -0.4512961
## med_high -0.36785679  0.1645982  0.15226017  0.4166923  0.09827811  0.4208230
## high     -0.48724019  1.0171306 -0.15538550  1.0045416 -0.47250601  0.8098159
##                 dis        rad        tax     ptratio      black       lstat
## low       0.8845325 -0.6886218 -0.7567361 -0.37768596  0.3791635 -0.78111842
## med_low   0.4035369 -0.5529539 -0.5132547 -0.01151131  0.3234892 -0.19001948
## med_high -0.3735685 -0.4053858 -0.2987757 -0.30822868  0.1283773 -0.01850998
## high     -0.8573713  1.6379981  1.5139626  0.78062517 -0.8596600  0.96934948
##                  medv
## low       0.537232159
## med_low   0.003866546
## med_high  0.161872943
## high     -0.740561820
## 
## Coefficients of linear discriminants:
##                 LD1         LD2         LD3
## zn       0.07739015  0.70485482 -0.94757111
## indus    0.03442359 -0.28585326  0.59355289
## chas    -0.09986824 -0.03341233  0.09490689
## nox      0.30265239 -0.73971012 -1.25654089
## rm      -0.10343239 -0.11362586 -0.05313860
## age      0.25622484 -0.35410860 -0.31036464
## dis     -0.08265281 -0.30651411  0.19906543
## rad      3.18424483  0.88896601  0.06549547
## tax      0.01258437 -0.05088693  0.15554547
## ptratio  0.08250346  0.12639893 -0.28012998
## black   -0.12817334 -0.02200587  0.10961806
## lstat    0.26121936 -0.16069426  0.43245625
## medv     0.19022990 -0.34641337 -0.21014934
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9487 0.0377 0.0136
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  graphics::arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda (bi)plot
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)

Task 6 - Testing the LDA for how well it could predict crime in Boston towns

# save the correct classes from test data
correct_classes <- test$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       14      10        0    0
##   med_low    4      13       11    0
##   med_high   0       7       16    1
##   high       0       0        0   26

\(~\) ##### Looking at model it performs well on the high crime prediction, but the low, med_low and med_high seem to be off slightly. med_low is the next best predicted crime level.

Task 7 - K means clusterring

data("Boston")
Boston_scaled <- as.data.frame(scale(Boston))
scaling the original data
dist_eu <- dist(Boston_scaled, method = "euclidean")
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970
measuring the euclidean distances between observations
#dist_man <- dist(Boston_scaled, method = "manhattan")
#dist_man
using the method = “manhattan” to shoud the absolute distances between the observations
set.seed(123)

# determine the number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

\(~\) ##### Looking at the elbow it appears that 2 is the optimal number of clusters for k means

# k-means clustering
km <- kmeans(Boston, centers = 2)

# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)

##### I have no idea how to interpret this-

Bonus

#Data
set.seed(32)
data("Boston")
boston_scaled <- scale(Boston) %>% as.data.frame()

#Set seven clusters:
km_bonus <- kmeans(Boston, centers = 8)
boston_scaled$cluster <- km_bonus$cluster

Boston_bonus <- lda(cluster ~ ., data = boston_scaled)

# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  graphics::arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(boston_scaled$cluster)

# plot the lda (bi)plot
plot(Boston_bonus, dimen =2)
lda.arrows(Boston_bonus, myscale = 1)

Boston_bonus
## Call:
## lda(cluster ~ ., data = boston_scaled)
## 
## Prior probabilities of groups:
##          1          2          3          4          5          6          7 
## 0.27075099 0.08695652 0.15217391 0.12252964 0.10869565 0.05335968 0.18379447 
##          8 
## 0.02173913 
## 
## Group means:
##         crim         zn      indus        chas        nox          rm
## 1  1.0097765 -0.4872402  1.0662784 -0.04242540  0.9959393 -0.39626518
## 2 -0.4112593  1.8232528 -1.0587789 -0.27232907 -1.2032247  0.32608884
## 3 -0.4103240  0.4669153 -0.6939781  0.08558914 -0.7573615  0.40725570
## 4 -0.3115695 -0.4872402  0.6973678  0.10868063  0.6498658 -0.24065169
## 5 -0.3906267 -0.1882696 -0.4589025 -0.05757815 -0.6526048  0.01656427
## 6 -0.4105500  0.7276120 -0.7660389 -0.27232907 -0.8389328  0.12107492
## 7 -0.3726011 -0.2488802 -0.5624188  0.15101504 -0.2307439  0.26980542
## 8 -0.1918628 -0.4872402  0.8121161  0.08558914  1.3206359 -0.52452958
##          age         dis        rad         tax    ptratio      black
## 1  0.7599946 -0.82659650  1.5757732  1.53915759  0.8040926 -0.7189340
## 2 -1.4995727  1.76550267 -0.6321108 -0.51640025 -0.7080115  0.3436424
## 3 -0.6511612  0.50825445 -0.6835681 -1.09300067 -0.2158123  0.3816348
## 4  0.7665217 -0.74688501 -0.5298939  0.01868990 -0.2268036  0.2441186
## 5 -1.0126162  0.40445826 -0.5725993 -0.71589771 -0.1533051  0.3386187
## 6 -1.1910518  0.90003854 -0.5735274 -0.05678564 -0.3404312  0.3375271
## 7  0.4650140  0.04164319 -0.5323637 -0.67571060 -0.2506439  0.3386363
## 8  0.8257272 -0.69874374 -0.5538062 -0.12654817 -0.6723188 -1.8525431
##        lstat       medv
## 1  0.8432167 -0.6807081
## 2 -0.8161977  0.3320129
## 3 -0.5638615  0.6660396
## 4  0.2594944 -0.1294832
## 5 -0.5252045  0.2809099
## 6 -0.5717207  0.2900036
## 7 -0.1610779  0.2011491
## 8  0.6385135 -0.5996044
## 
## Coefficients of linear discriminants:
##                  LD1          LD2         LD3         LD4          LD5
## crim     0.009913887 -0.137609161 -0.04036563  0.06710828  0.008706277
## zn      -0.063869885 -0.359539354 -0.03252327 -0.37386866 -0.985470185
## indus   -0.064553782 -0.195044951 -0.33201434  0.87866138 -0.853101757
## chas     0.051357463  0.035927002 -0.01879004 -0.06102304  0.054091077
## nox     -0.064270060  0.327990930  0.43933045  0.36992087 -0.682424547
## rm      -0.031831393  0.027532509 -0.07621914 -0.14238990  0.106374027
## age     -0.470229341  1.450265293 -0.23689536 -1.46202224 -0.271967600
## dis      0.361412866 -0.313512208  0.20731668 -0.62537375 -0.395082557
## rad     -1.643884814 -0.791845574 -3.25137506 -0.13663787 -0.481829789
## tax     -9.099980888 -0.327782726  3.36875714 -0.64786007  1.410249515
## ptratio -0.008405428  0.060541663 -0.29843909  0.19223768 -0.102382405
## black    0.023126886 -0.089392096 -0.06586979 -0.19721961  0.616762841
## lstat   -0.032547271 -0.183432824 -0.09333305  0.15316806  0.091791088
## medv    -0.062060374  0.005458806 -0.02837714  0.34842936 -0.056421542
##                  LD6         LD7
## crim     0.098841166  0.03527449
## zn       0.842438878  0.35861794
## indus    0.904577426 -0.95614725
## chas    -0.048494403  0.02316471
## nox     -0.321696150  0.19808743
## rm       0.079509917 -0.04629482
## age      0.130244689 -0.20378571
## dis     -0.476692936 -0.65750487
## rad      0.001318226 -1.00217642
## tax     -0.281439177  0.85946009
## ptratio  0.336896632  0.79163091
## black    0.782771708 -0.43714271
## lstat    0.104232162  0.73179612
## medv     0.242827431  1.10579041
## 
## Proportion of trace:
##    LD1    LD2    LD3    LD4    LD5    LD6    LD7 
## 0.9703 0.0155 0.0084 0.0029 0.0018 0.0009 0.0002
it seams that tax and rad have a high influence on crime followed by age

####Super Bonus

model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
library(plotly)
## Warning: package 'plotly' was built under R version 4.3.2
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:reshape':
## 
##     rename
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers',color = train$crime)